The No-Curvature Interpretation of General Relativity

There is an interesting analogy between the theories of quantum mechanics and general relativity involving apparent non-linearity. Both theories may be considered to be based on a purely linear foundation, represented by the Schrödinger wave equation for quantum mechanics and the flat Minkowski spacetime metric for general relativity. For quantum mechanics we then traditionally imagine something like a "collapse of the wave function" when a measurement takes place, and this operation introduces non-linearity to the theory. Analogously in general relativity we imagine "curvature" of the observational spacetime manifold in accord with the non-linear Einstein field equations.

In the case of quantum mechanics there are alternative interpretations that do not explicitly involve a collapse of the wave-function, and that assume instead that the overall wave function always continues to evolve in accord with the purely linear Schrödinger equation. These are sometimes called "no-collapse" interpretations. According to some variants of this approach, we are asked to imagine a "branching" of the wave function into multiple alternate histories, leading to the so-called "many worlds" or "many histories" interpretations of quantum mechanics. Of course, it's necessary for these theories to somehow account for the apparent collapse and non-linearity in the course of events. In other words, they must explain why we seem to observe only a single world with a single history. For this purpose, the idea of "decoherence" is sometimes invoked.

Much less well-known is a somewhat analogous approach to general relativity, which we might call the "no-curvature interpretation”. This also involves "branching" into multiple layers while maintaining purely linear local evolution of each layer. To understand how this approach yields the effects of curvature while always dealing in terms of flat manifolds, it's useful to consider a simple two-dimensional surface. For any such surface, curved or not, we can partition the surface into a network of triangles, each sufficiently small so that the surface can be regarded as flat over an individual triangle.

The curvature of the surface arises from how these flat triangles fit together, just like the triangles used in surveying hilly terrain. As an example, if we find six equilateral triangles meeting at a given vertex, then the surface is flat at that vertex, whereas if we find only five equilateral triangles meeting at a vertex, it is curved, congruent to a vertex of an icosahedron. Notice that we referring here to intrinsic Gaussian curvature, and there is no intrinsic curvature along the edge between two faces. We can always flatten out two faces meeting along a straight edge, showing that the surface is metrically flat across such boundaries. All the curvature of the surface is concentrated at singularities at the vertices, as shown by the Gauss-Bonnet theorem for loops surrounding various point on the surface. Only loops circling vertices can reveal any curvature. (This is an interesting illustration of how "curvature" can be regarded as a non-local attribute, since a path can reveal curvature topologically even though the local curvature is everywhere zero on the path.)

But suppose that, instead of attaching all five of those equilateral triangles together along their edges, we branch into a new layer of the surface, so we are continuing to circle the vertex, not returning to the original face after the first circuit, but arriving at a face coinciding with the first but on a different layer. We can then continue to circle the vertex and never implicate any metrical curvature of the actual multi-sheeted manifold. For the specific case of an icosahedral vertex we can actually close the manifold around the vertex by joining the edges after six windings in the icosahedral interpretation, which corresponds to five windings in the flat interpretation. (For a more detailed discussion, see the note on The Tetratorus.)

The simplest illustration of a complete closed surface with two alternative interpretations is the tetratorus, which can be described as either a two-layered surface in the shape of a sphere (with positive intrinsic curvature), or a one-layered torroidal surface (with no curvature). Incidentally, for those familiar with complex analytic functions, this corresponds exactly to the Riemann surface of the two-valued function sqrt[p(z)] for a cubic polynomial p with distinct roots, which gives a two-sheeted sphere homeomorphic to a one-sheeted torus.

In general it is always possible to interpret any arbitrarily curved surface, closed or open, with any topology, as an intrinsically flat surface wrapped into suitably connected layers. Of course, this implies that each face of the interpreted surface actually represents multiple layers, and the "curvature" emerges from identifying these layers modulo the interpreted shape.

For purposes of expressing a physical theory of spacetime, we now face a task similar to that in quantum mechanics of explaining the physical and phenomenological significance of the multiple linear branches, and accounting for the appearance of just a single branch. One trivial approach would be to simply stipulate empirical indistinguishability of the layers, which would make the entire construction merely conventional. In other words, it serves as an un-defeatable example in support of Poincare's thesis that geometry is conventional, and in particular the fact that we can always conceive of a curved surface as a flat one, provided we are willing to conceptually decompose the surface into layers which may be empirically indistinguishable.

Another, more interesting, approach would be take seriously the idea of multiple sheets, regarding the spacetime manifold as something like the Riemann surface (in the sense of analytic function theory) of a multi-valued function whose zeros correspond to the vertices (particles?) of the manifold. It's worth noting that the quantum wave functions of some particles have the property that they are returned to their original state not by a rotation of 360 degrees but by "going around twice", i.e., by a rotation of 720 degrees. This might lead us to associate multiple sheets of spacetime around a vertex with some aspect of the quantum wave function of a particle.

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